148 Chapter 5Integration
5.6 Static properties of matter
Consider a set of Ndiscrete massesm
1,m
2, =,m
Ndistributed along a (horizontal)
straight line, with massm
iat positionx
iwith respect to a fixed point Oas in Figure 5.15.
The system of masses has the following properties:
(i) total mass:
(ii) first moment of mass about O:
(iii) second moment of mass about O: (5.39)
and there are similar definitions for all higher moments.
The first moment of mass, (ii), defines the position Xof the centre of mass of the
system of masses:
(5.40)
The first moment of the system of masses is therefore equal to the moment of the total
mass Mconcentrated at the centre of mass.
In the presence of gravity, each mass experiences a force F
i1 = 1 m
igdirected
downwards (Figure 5.16), where gis the standard acceleration of gravity (‘gravitational
constant’). This force F
iis what is commonly called the weight of the body.
Multiplication of equation (5.40) by ggives
(5.41)
TFxMgXFXiNii===
=∑
1iNiimx MX
=∑
=
1iNiimx
=∑
12iNiimx
=∑
1Mm
iiN=
=∑
1.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................••• x ••
m
1m
2m
3m
N− 1m
Nx
1x
2x
3 ··· ···
x
N− 1x
No
x
............
.......
........
...........................Figure 5.15
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...................................F
1F
2F
3F
N− 1F
Nx
1x
2x
3 ··· ···
x
N− 1x
No
x
................
.......
........
.......................Figure 5.16